Properties

Degree 4
Conductor 277
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3-s − 5-s + 2·6-s + 7-s + 4·8-s + 2·10-s − 2·11-s + 3·13-s − 2·14-s + 15-s − 4·16-s − 4·17-s − 19-s − 21-s + 4·22-s + 3·23-s − 4·24-s + 3·25-s − 6·26-s − 2·27-s − 29-s − 2·30-s − 10·31-s + 2·33-s + 8·34-s − 35-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s − 0.447·5-s + 0.816·6-s + 0.377·7-s + 1.41·8-s + 0.632·10-s − 0.603·11-s + 0.832·13-s − 0.534·14-s + 0.258·15-s − 16-s − 0.970·17-s − 0.229·19-s − 0.218·21-s + 0.852·22-s + 0.625·23-s − 0.816·24-s + 3/5·25-s − 1.17·26-s − 0.384·27-s − 0.185·29-s − 0.365·30-s − 1.79·31-s + 0.348·33-s + 1.37·34-s − 0.169·35-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(277\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{277} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 277,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.1431366605$
$L(\frac12)$  $\approx$  $0.1431366605$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 277$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 277$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad277$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 8 T + p T^{2} ) \)
good2$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \)
3$D_{4}$ \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 3 T + 7 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + T + 13 T^{2} + p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 10 T + 68 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$D_{4}$ \( 1 - 7 T + 37 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 36 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$D_{4}$ \( 1 - 14 T + 136 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - T + 43 T^{2} - p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$D_{4}$ \( 1 - 5 T - 10 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 - 2 T + 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.8383227474, −19.1745567864, −18.6003292006, −18.0126226421, −17.8685107407, −17.0212695469, −16.5112504108, −15.8575391568, −15.0261185798, −14.2430115276, −13.1187710094, −12.9354901068, −11.4860616488, −11.0355703521, −10.3714949973, −9.31193442615, −8.74527851838, −8.01150574486, −7.07871685645, −5.58349348623, −4.30532032866, 4.30532032866, 5.58349348623, 7.07871685645, 8.01150574486, 8.74527851838, 9.31193442615, 10.3714949973, 11.0355703521, 11.4860616488, 12.9354901068, 13.1187710094, 14.2430115276, 15.0261185798, 15.8575391568, 16.5112504108, 17.0212695469, 17.8685107407, 18.0126226421, 18.6003292006, 19.1745567864, 19.8383227474

Graph of the $Z$-function along the critical line